To solve this problem, we need to calculate the standard error (SE) and then find the range of proportions based on the given formula. Here’s a step-by-step explanation:

Given:

• Null hypothesis: $H_0: p = 0.05$
• Sample size: $n = 2000$
• Confidence level: 95%, which corresponds to $1.96$ standard deviations from the mean in a normal distribution.

Step 1: Calculate the Standard Error (SE)

The formula for the standard error when estimating a proportion is: $SE = \sqrt{\frac{p(1-p)}{n}}$

Substitute $p = 0.05$ and $n = 2000$: $SE = \sqrt{\frac{0.05(1-0.05)}{2000}}$ $SE = \sqrt{\frac{0.05 \times 0.95}{2000}}$ $SE = \sqrt{\frac{0.0475}{2000}}$ $SE \approx \sqrt{0.00002375} \approx 0.00487$

Step 2: Calculate the 95% Confidence Interval

Now use the formula $p \pm 1.96 \times SE$: $0.05 \pm 1.96 \times 0.00487$ $0.05 \pm 0.00954$

This gives us the range: $0.04046 \text{ to } 0.05954$

Therefore, if the null hypothesis is true, we would expect to find the proportion of NOVIDs in the range of approximately 4.05% to 5.95% with a probability of 95%.

What to Study to Understand This Concept Better:

1. Basic Probability Theory: Understanding probability distributions, particularly the normal distribution.
2. Statistical Hypothesis Testing: Learning about null and alternative hypotheses, Type I and Type II errors.
3. Confidence Intervals: Understanding the concept of confidence intervals and their interpretation in statistical analysis.
4. Sampling and Standard Error: Learning how the standard error is calculated and why it is used in statistical estimation.
5. Z-scores and the Standard Normal Distribution: Knowing how to use z-scores to determine probabilities and critical values in hypothesis testing.

Additional Questions:

1. What would the confidence interval look like for a different sample size, such as 1000?
2. How does increasing the confidence level to 99% affect the confidence interval range?
3. What happens to the standard error if the proportion $p$ is closer to 0.5?
4. Why do we use $1.96$ as the multiplier for a 95% confidence interval?
5. How would we interpret the results if the observed proportion is outside the calculated confidence interval?

Tip:

When dealing with proportions, the larger the sample size, the smaller the standard error, leading to a narrower confidence interval for the same confidence level.